Use the following information for the problem.

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

State of Probability of Returns If State of Economy Occurs

Economy State of Economy Stock S Stock T

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

Boom $0\mathrm{.}200\mathrm{.}20($or $20\%)0\mathrm{.}05($or $5\%)$

Normal $0\mathrm{.}500\mathrm{.}10($or $10\%)0\mathrm{.}07($or $7\%)$

Recession $0\mathrm{.}300\mathrm{.}05($or $5\%)0\mathrm{.}15($or $15\%)\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

a$)$ Find the expected return of each stock.

Use at least seven decimal places in computations of $($b$),($c$)$ and $($d$)$ below to avoid significant rounding errors.

b$)$ Calculate the variance and standard deviation of returns of each stock.

c$)$ Compute the covariance and correlation of returns between the two stocks.

d$)$ Assume that you invest $$40,000$ in Stock S and $$60,000$ in Stock T$.$ Find the expected return on the portfolio and the standard deviation of the portfolio$$s return.